Frustrated Cooper pairing and $f$-wave supersolidity in cold-atom optical lattices

Geometric frustration in quantum magnetism refers to the fact that magnetic interactions on different bonds cannot be simultaneously minimized. The usual Cooper pairing systems favor uniform spatial distributions of pairing phases among different lattice sites without frustration. In contrast, we propose “frustrated Cooper pairing” in non-bipartite lattices which leads to supersolid states of Cooper pairs. Not only the amplitudes of the pairing order parameter but also its signs vary from site to site. This exotic pairing state naturally occurs in the $p$-orbital bands in optical lattices with ultracold spinless fermions. In the triangular lattice, it exhibits an unconventional supersolid state with the $f$-wave symmetry.

Figure 1

(a) The $\sigma$ bonding and $\pi$ bonding of the $p$ orbitals have opposite signs due to their odd parity nature. This gives rise to the antiferromagnetic-like exchange in the change channel with attractive interactions as expressed in Eq. (8), which is frustrated in the triangular lattice. (b) The first Brillouin zone of the triangular lattice is a regular hexagon. $K_{1,2}=(\pm \frac{4\pi }{3a},0)$ represent two nonequivalent vertices, and the other four are equivalent to $K_{1,2}$.

Figure 2

The band structure of the Hamiltonian Eq. (1). Two bands touch at $K_{1,2}$ with the Dirac spectra, and at the center of the BZ with the quadratic spectra.

Figure 3

The pattern of $\langle G\left|\mathrm{\eta ⃗}\right|G\rangle$ at $h=0$: The unit cell of three sites exhibiting supersolid ordering.

Figure 4

The average fermion number per site $n$ versus the chemical potential $\mu$ at $U/t_{\parallel }=6$ and $t_{\perp }/t_{\parallel }=0.2$.

Figure 5

The real space configurations of pseudospin $\mathrm{\eta ⃗}$ on the $\mathit{xz}$ plane at various fillings $n$ from (a) to (i). (a) and (i) indicate fully polarized states. (b) and (h) show three titled vectors, where two of them have a relative $\pi$ phase to the third one. (c) and (g) depict the CDW insulating state. (d) and (f) exhibit an umbrella-like shape with opposite orientation. Two of them have a $\pi$-phase difference and the third one does not have a superfluid component. (e) denotes an intermediate configuration between (d) and (f).

Figure 6

The smooth evolution of the two opposite umbrella-like configurations from $n=0.68$ to $n=1$.

Figure 7

The $f$-wave pairing pattern in (a) real space and (b) momentum space. The rotation of ${60}^{°}$ around the center site $O$ in (a) (denoted by the hollow circle) in real space or around the center of the reduced BZ in momentum space is equivalent to reversing the sign of the pairing order parameters.

Figure 8

(a) The reduced Brillouin zone (RBZ) associated with the enlarged three-site unit cell compared with the original BZ. The six vertices of the RBZ are located at the centers of the six regular triangles composed of the center of the BZ and the vertices of the original BZ. (b) The six bands in the reduced BZ with the three-site CDW pattern of $n_A=n_C=1.52$, and $n_B=0.154$ for $t_{\perp }=0.2$ and $U=6$. The chemical potential $\mu$ is reset to zero.